We study the family of presistances on graphs for p ≥ 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p=1, the presistance coincides with the shortest path distance, for p=2 it coincides with the standard resistance distance, and for p → ∞ it converges to the inverse of the minimal stcut in the graph. Secondly, we consider the special case of random geometric graphs (such as knearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phasetransition takes place. There exist two critical thresholds p^* and p^** such that if p < p^*, then the presistance depends on meaningful global properties of the graph, whereas if p > p^**, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p^* = 1 + 1/(d1) and p^** = 1 + 1/(d2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p^* and p^** is an artifact of our proofs. We also relate our findings to Laplacian regularization and suggest to use qLaplacians as regularizers, where q satisfies 1/p^* + 1/q = 1.
Author(s):  Alamgir, M. and von Luxburg, U. 
Book Title:  Advances in Neural Information Processing Systems 24 
Pages:  379387 
Year:  2011 
Day:  0 
Editors:  J ShaweTaylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger 
Department(s):  Empirical Inference 
Research Project(s): 
Random geometric graphs

Bibtex Type:  Conference Paper (inproceedings) 
Event Name:  TwentyFifth Annual Conference on Neural Information Processing Systems (NIPS 2011) 
Event Place:  Granada, Spain 
Digital:  0 
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BibTex @inproceedings{Alamgirv2011, title = {Phase transition in the family of presistances}, author = {Alamgir, M. and von Luxburg, U.}, booktitle = {Advances in Neural Information Processing Systems 24}, pages = {379387}, editors = {J ShaweTaylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger}, year = {2011} } 